Measure, Integration & Real Analysis · Problem 6E.5

L^p Spaces — Sheldon Axler

Mathematics — Measure Theory · complete-metric-space-baire

Problem

Suppose

X = \{0\} \cup \bigcup_{k=1}^{\infty} \left\{\tfrac{1}{k}\right\}

and

d(x,y) = |x - y|

for

x, y \in X

. (a) Show that

(X, d)

is a complete metric space. (b) Each set of the form

\{x\}

for

x \in X

is a closed subset of

\mathbf{R}

that has an empty interior as a subset of

\mathbf{R}

. Clearly

X

is a countable union of such sets. Explain why this does not violate the statement of Baire's Theorem that a complete metric space is not the countable union of closed subsets with empty interior.

Related theory

Metric Space Completeness And Baire Category Theorem

Browse the problem library